In `DeltaPQR`,the measure of angle Q is `90^(@)`.If `cosP=(5)/(13),andPQ=2.4cm` , what is the length (in cm .)of side PR?

Options

A. 6
B. 6.5
C. 4
D. 5.6

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the triangle and given values
We have a right triangle \( \Delta PQR \) where \( \angle Q = 90^\circ \). We know that \( \cos P = \frac{5}{13} \) and \( PQ = 2.4 \, \text{cm} \).

**Hint:** Remember that in a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

### Step 2: Identify the sides based on cosine
From the definition of cosine, we have:
\[ \cos P = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{PQ}{PR} \]
Given \( \cos P = \frac{5}{13} \), we can set:
- Adjacent side (to angle P) = \( PQ = 5k \)
- Hypotenuse = \( PR = 13k \)

**Hint:** Here, \( k \) is a scaling factor that will help us relate the actual lengths to the ratios.

### Step 3: Set up the equation for \( PQ \)
We know \( PQ = 2.4 \, \text{cm} \). From our earlier step, we have:
\[ PQ = 5k \]
Thus,
\[ 5k = 2.4 \]

**Hint:** To find \( k \), divide both sides by 5.

### Step 4: Solve for \( k \)
\[ k = \frac{2.4}{5} = 0.48 \]

**Hint:** This value of \( k \) will help us find the lengths of the other sides.

### Step 5: Find the length of \( PR \)
Now, we can find \( PR \) using the hypotenuse formula:
\[ PR = 13k = 13 \times 0.48 \]

**Hint:** Multiply to find the actual length of \( PR \).

### Step 6: Calculate \( PR \)
\[ PR = 13 \times 0.48 = 6.24 \, \text{cm} \]

**Hint:** Ensure your multiplication is accurate to arrive at the correct length.

### Conclusion
The length of side \( PR \) is \( 6.24 \, \text{cm} \).

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